I will explain how to associate a version of the KP hierarchy to the oriented cyclic quiver with m vertices. Rational solutions of that hierarchy are parametrized by the points of suitable quiver varieties. The pole dynamics of the solutions is linked to the Calogero-Moser problem for the complex reflection group G(m,1,n). This can be viewed as a generalisation of the well-known result of George Wilson about the usual KP hierarchy (corresponding to m=1).
This is joint work with Alexey Silantyev (Leeds).
Quantum cluster algebras are a natural noncommutative generalisation of cluster algebras. This noncommutativity is relatively mild: the main feature is that elements in the same cluster must commute up to a power of q, although elements from different clusters may have more complicated relationships. Many results for cluster algebras have direct parallels for the quantum case and we will discuss some of what is now known and not known for quantum cluster algebras.
We will also study gradings, show how they can be classified and use them to produce twisted cluster algebra structures. On the one hand, gradings bring out some beautiful combinatorics, in the form of frieze patterns, but we will also consider a more algebraic application of gradings, namely a "twisting" construction that is important for quantizing cluster algebra structures.
We introduce a class of supercommutative superalgebras generalizing the cluster algebras of Fomin and Zelevinsky. Our main examples are supersymmetric analogs of the classical Coxeter frieze patterns. The notion of "superfrieze" is related with linear difference operators called supersymmetric Hill's operators.
There will be presented new current algebras on Riemann surfaces, Lax operator algebras by name. In particular, this class of Lie algebras contains the (non-twisted) Kac--Moody algebras. Based on this approach, an outline of the theory of finite-dimensional Lax integrable systems with spectral parameter on a Riemann surface will be given. It includes the systems with rational spectral parameter as well. The list of examples includes Calogero-Moser and Hitchin systems, classical giroscopes and integrable cases of flow around a rigid body. We shall formulate a general ansatz for the operators of Lax pairs, construct hirearchies of commuting flows, and their Hamiltonians in terms of semi-simple Lie algebra invariants. As examples, we will show the Lax pairs for the elliptic Calogero-Moser systems for classical Lie algebras, calculate number of independent Hamiltonians for the Hitchin systems.
In this talk, we show the indecomposable sl(2,C) modules in the Bernstein-Gelfand-Gelfand (BGG) category O naturally arise for homogeneous integrable nonlinear evolutionary systems. We then develop an approach to construct master symmetries for such integrable systems. This method enables us to compute the hierarchy of time-dependent symmetries. We finally illustrate the method using both classical and new examples. We compare our approach to the known existing methods used to construct master symmetries. For the new integrable equations such as a Benjamin-Ono type equation, a new integrable Davey-Stewartson type equation and two different versions of (2+1)-dimensional generalised Volterra Chains, we generate their conserved densities using their master symmetries.
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Last updated 15th May, 2015